I was wondering what is the fastest way to solve $X^2-1801Y^2=1$ and how many iterations would it take by such method? The solution is humongous (the $Y$ solution apparently has 56 digits) and I am dying with the continued fraction method. Can someone please help?
Thanks
This is a variant to continued fractions due to Gauss and Lagrange. Since 1801 is a prime 1 mod 4, there is a solution to $u^2 - 1801 v^2 = -1.$ Check form number 63 in the output. Then $$ (u^2 + 1801 v^2)^2 - 1801 (2uv)^2 = 1 $$
Anyway, only integer operations, no approximations used... I added commands to print the time at beginning and end of the computation. This takes under one second.
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./Pell 1801
Sun Jul 14 17:06:47 PDT 2019
0 form 1 84 -37 delta -2
1 form -37 64 21 delta 3
2 form 21 62 -40 delta -1
3 form -40 18 43 delta 1
4 form 43 68 -15 delta -5
5 form -15 82 8 delta 10
6 form 8 78 -35 delta -2
7 form -35 62 24 delta 3
8 form 24 82 -5 delta -16
9 form -5 78 56 delta 1
10 form 56 34 -27 delta -2
11 form -27 74 16 delta 4
12 form 16 54 -67 delta -1
13 form -67 80 3 delta 27
14 form 3 82 -40 delta -2
15 form -40 78 7 delta 11
16 form 7 76 -51 delta -1
17 form -51 26 32 delta 1
18 form 32 38 -45 delta -1
19 form -45 52 25 delta 2
20 form 25 48 -49 delta -1
21 form -49 50 24 delta 2
22 form 24 46 -53 delta -1
23 form -53 60 17 delta 4
24 form 17 76 -21 delta -3
25 form -21 50 56 delta 1
26 form 56 62 -15 delta -4
27 form -15 58 64 delta 1
28 form 64 70 -9 delta -8
29 form -9 74 48 delta 1
30 form 48 22 -35 delta -1
31 form -35 48 35 delta 1
32 form 35 22 -48 delta -1
33 form -48 74 9 delta 8
34 form 9 70 -64 delta -1
35 form -64 58 15 delta 4
36 form 15 62 -56 delta -1
37 form -56 50 21 delta 3
38 form 21 76 -17 delta -4
39 form -17 60 53 delta 1
40 form 53 46 -24 delta -2
41 form -24 50 49 delta 1
42 form 49 48 -25 delta -2
43 form -25 52 45 delta 1
44 form 45 38 -32 delta -1
45 form -32 26 51 delta 1
46 form 51 76 -7 delta -11
47 form -7 78 40 delta 2
48 form 40 82 -3 delta -27
49 form -3 80 67 delta 1
50 form 67 54 -16 delta -4
51 form -16 74 27 delta 2
52 form 27 34 -56 delta -1
53 form -56 78 5 delta 16
54 form 5 82 -24 delta -3
55 form -24 62 35 delta 2
56 form 35 78 -8 delta -10
57 form -8 82 15 delta 5
58 form 15 68 -43 delta -1
59 form -43 18 40 delta 1
60 form 40 62 -21 delta -3
61 form -21 64 37 delta 2
62 form 37 84 -1 delta -84
63 form -1 84 37 delta 2
64 form 37 64 -21 delta -3
65 form -21 62 40 delta 1
66 form 40 18 -43 delta -1
67 form -43 68 15 delta 5
68 form 15 82 -8 delta -10
69 form -8 78 35 delta 2
70 form 35 62 -24 delta -3
71 form -24 82 5 delta 16
72 form 5 78 -56 delta -1
73 form -56 34 27 delta 2
74 form 27 74 -16 delta -4
75 form -16 54 67 delta 1
76 form 67 80 -3 delta -27
77 form -3 82 40 delta 2
78 form 40 78 -7 delta -11
79 form -7 76 51 delta 1
80 form 51 26 -32 delta -1
81 form -32 38 45 delta 1
82 form 45 52 -25 delta -2
83 form -25 48 49 delta 1
84 form 49 50 -24 delta -2
85 form -24 46 53 delta 1
86 form 53 60 -17 delta -4
87 form -17 76 21 delta 3
88 form 21 50 -56 delta -1
89 form -56 62 15 delta 4
90 form 15 58 -64 delta -1
91 form -64 70 9 delta 8
92 form 9 74 -48 delta -1
93 form -48 22 35 delta 1
94 form 35 48 -35 delta -1
95 form -35 22 48 delta 1
96 form 48 74 -9 delta -8
97 form -9 70 64 delta 1
98 form 64 58 -15 delta -4
99 form -15 62 56 delta 1
100 form 56 50 -21 delta -3
101 form -21 76 17 delta 4
102 form 17 60 -53 delta -1
103 form -53 46 24 delta 2
104 form 24 50 -49 delta -1
105 form -49 48 25 delta 2
106 form 25 52 -45 delta -1
107 form -45 38 32 delta 1
108 form 32 26 -51 delta -1
109 form -51 76 7 delta 11
110 form 7 78 -40 delta -2
111 form -40 82 3 delta 27
112 form 3 80 -67 delta -1
113 form -67 54 16 delta 4
114 form 16 74 -27 delta -2
115 form -27 34 56 delta 1
116 form 56 78 -5 delta -16
117 form -5 82 24 delta 3
118 form 24 62 -35 delta -2
119 form -35 78 8 delta 10
120 form 8 82 -15 delta -5
121 form -15 68 43 delta 1
122 form 43 18 -40 delta -1
123 form -40 62 21 delta 3
124 form 21 64 -37 delta -2
125 form -37 84 1 delta 84
126 form 1 84 -37
disc 7204
Automorph, written on right of Gram matrix:
8329143980792153679548730627982433105816857717738527006289 703297844884742717748143927348581545599367975438677758224360
19008049861749803182382268306718420151334269606450750222280 1605005332367775620999659268392329725817895504659601545677809
Pell automorph
806667238174283887339603999510156079461856181188670036342049 34233497801011395531470465220399874692553019561217801150326280
19008049861749803182382268306718420151334269606450750222280 806667238174283887339603999510156079461856181188670036342049
Pell unit
806667238174283887339603999510156079461856181188670036342049^2 - 1801 * 19008049861749803182382268306718420151334269606450750222280^2 = 1
=========================================
Pell NEGATIVE
635085521081328025318961532468^2 - 1801 * 14964952932154520840080376605^2 = -1
=========================================
1801 1801
Sun Jul 14 17:06:47 PDT 2019
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$
just the part where the "digits" are calculated:
Method described by Prof. Lubin at Continued fraction of $\sqrt{67} - 4$
$$ \sqrt { 1801} = 42 + \frac{ \sqrt {1801} - 42 }{ 1 } $$ $$ \frac{ 1 }{ \sqrt {1801} - 42 } = \frac{ \sqrt {1801} + 42 }{37 } = 2 + \frac{ \sqrt {1801} - 32 }{37 } $$ $$ \frac{ 37 }{ \sqrt {1801} - 32 } = \frac{ \sqrt {1801} + 32 }{21 } = 3 + \frac{ \sqrt {1801} - 31 }{21 } $$ $$ \frac{ 21 }{ \sqrt {1801} - 31 } = \frac{ \sqrt {1801} + 31 }{40 } = 1 + \frac{ \sqrt {1801} - 9 }{40 } $$ $$ \frac{ 40 }{ \sqrt {1801} - 9 } = \frac{ \sqrt {1801} + 9 }{43 } = 1 + \frac{ \sqrt {1801} - 34 }{43 } $$ $$ \frac{ 43 }{ \sqrt {1801} - 34 } = \frac{ \sqrt {1801} + 34 }{15 } = 5 + \frac{ \sqrt {1801} - 41 }{15 } $$ $$ \frac{ 15 }{ \sqrt {1801} - 41 } = \frac{ \sqrt {1801} + 41 }{8 } = 10 + \frac{ \sqrt {1801} - 39 }{8 } $$ $$ \frac{ 8 }{ \sqrt {1801} - 39 } = \frac{ \sqrt {1801} + 39 }{35 } = 2 + \frac{ \sqrt {1801} - 31 }{35 } $$ $$ \frac{ 35 }{ \sqrt {1801} - 31 } = \frac{ \sqrt {1801} + 31 }{24 } = 3 + \frac{ \sqrt {1801} - 41 }{24 } $$ $$ \frac{ 24 }{ \sqrt {1801} - 41 } = \frac{ \sqrt {1801} + 41 }{5 } = 16 + \frac{ \sqrt {1801} - 39 }{5 } $$ $$ \frac{ 5 }{ \sqrt {1801} - 39 } = \frac{ \sqrt {1801} + 39 }{56 } = 1 + \frac{ \sqrt {1801} - 17 }{56 } $$ $$ \frac{ 56 }{ \sqrt {1801} - 17 } = \frac{ \sqrt {1801} + 17 }{27 } = 2 + \frac{ \sqrt {1801} - 37 }{27 } $$ $$ \frac{ 27 }{ \sqrt {1801} - 37 } = \frac{ \sqrt {1801} + 37 }{16 } = 4 + \frac{ \sqrt {1801} - 27 }{16 } $$ $$ \frac{ 16 }{ \sqrt {1801} - 27 } = \frac{ \sqrt {1801} + 27 }{67 } = 1 + \frac{ \sqrt {1801} - 40 }{67 } $$ $$ \frac{ 67 }{ \sqrt {1801} - 40 } = \frac{ \sqrt {1801} + 40 }{3 } = 27 + \frac{ \sqrt {1801} - 41 }{3 } $$ $$ \frac{ 3 }{ \sqrt {1801} - 41 } = \frac{ \sqrt {1801} + 41 }{40 } = 2 + \frac{ \sqrt {1801} - 39 }{40 } $$ $$ \frac{ 40 }{ \sqrt {1801} - 39 } = \frac{ \sqrt {1801} + 39 }{7 } = 11 + \frac{ \sqrt {1801} - 38 }{7 } $$ $$ \frac{ 7 }{ \sqrt {1801} - 38 } = \frac{ \sqrt {1801} + 38 }{51 } = 1 + \frac{ \sqrt {1801} - 13 }{51 } $$ $$ \frac{ 51 }{ \sqrt {1801} - 13 } = \frac{ \sqrt {1801} + 13 }{32 } = 1 + \frac{ \sqrt {1801} - 19 }{32 } $$ $$ \frac{ 32 }{ \sqrt {1801} - 19 } = \frac{ \sqrt {1801} + 19 }{45 } = 1 + \frac{ \sqrt {1801} - 26 }{45 } $$ $$ \frac{ 45 }{ \sqrt {1801} - 26 } = \frac{ \sqrt {1801} + 26 }{25 } = 2 + \frac{ \sqrt {1801} - 24 }{25 } $$ $$ \frac{ 25 }{ \sqrt {1801} - 24 } = \frac{ \sqrt {1801} + 24 }{49 } = 1 + \frac{ \sqrt {1801} - 25 }{49 } $$ $$ \frac{ 49 }{ \sqrt {1801} - 25 } = \frac{ \sqrt {1801} + 25 }{24 } = 2 + \frac{ \sqrt {1801} - 23 }{24 } $$ $$ \frac{ 24 }{ \sqrt {1801} - 23 } = \frac{ \sqrt {1801} + 23 }{53 } = 1 + \frac{ \sqrt {1801} - 30 }{53 } $$ $$ \frac{ 53 }{ \sqrt {1801} - 30 } = \frac{ \sqrt {1801} + 30 }{17 } = 4 + \frac{ \sqrt {1801} - 38 }{17 } $$ $$ \frac{ 17 }{ \sqrt {1801} - 38 } = \frac{ \sqrt {1801} + 38 }{21 } = 3 + \frac{ \sqrt {1801} - 25 }{21 } $$ $$ \frac{ 21 }{ \sqrt {1801} - 25 } = \frac{ \sqrt {1801} + 25 }{56 } = 1 + \frac{ \sqrt {1801} - 31 }{56 } $$ $$ \frac{ 56 }{ \sqrt {1801} - 31 } = \frac{ \sqrt {1801} + 31 }{15 } = 4 + \frac{ \sqrt {1801} - 29 }{15 } $$ $$ \frac{ 15 }{ \sqrt {1801} - 29 } = \frac{ \sqrt {1801} + 29 }{64 } = 1 + \frac{ \sqrt {1801} - 35 }{64 } $$ $$ \frac{ 64 }{ \sqrt {1801} - 35 } = \frac{ \sqrt {1801} + 35 }{9 } = 8 + \frac{ \sqrt {1801} - 37 }{9 } $$ $$ \frac{ 9 }{ \sqrt {1801} - 37 } = \frac{ \sqrt {1801} + 37 }{48 } = 1 + \frac{ \sqrt {1801} - 11 }{48 } $$ $$ \frac{ 48 }{ \sqrt {1801} - 11 } = \frac{ \sqrt {1801} + 11 }{35 } = 1 + \frac{ \sqrt {1801} - 24 }{35 } $$ $$ \frac{ 35 }{ \sqrt {1801} - 24 } = \frac{ \sqrt {1801} + 24 }{35 } = 1 + \frac{ \sqrt {1801} - 11 }{35 } $$ $$ \frac{ 35 }{ \sqrt {1801} - 11 } = \frac{ \sqrt {1801} + 11 }{48 } = 1 + \frac{ \sqrt {1801} - 37 }{48 } $$ $$ \frac{ 48 }{ \sqrt {1801} - 37 } = \frac{ \sqrt {1801} + 37 }{9 } = 8 + \frac{ \sqrt {1801} - 35 }{9 } $$ $$ \frac{ 9 }{ \sqrt {1801} - 35 } = \frac{ \sqrt {1801} + 35 }{64 } = 1 + \frac{ \sqrt {1801} - 29 }{64 } $$ $$ \frac{ 64 }{ \sqrt {1801} - 29 } = \frac{ \sqrt {1801} + 29 }{15 } = 4 + \frac{ \sqrt {1801} - 31 }{15 } $$ $$ \frac{ 15 }{ \sqrt {1801} - 31 } = \frac{ \sqrt {1801} + 31 }{56 } = 1 + \frac{ \sqrt {1801} - 25 }{56 } $$ $$ \frac{ 56 }{ \sqrt {1801} - 25 } = \frac{ \sqrt {1801} + 25 }{21 } = 3 + \frac{ \sqrt {1801} - 38 }{21 } $$ $$ \frac{ 21 }{ \sqrt {1801} - 38 } = \frac{ \sqrt {1801} + 38 }{17 } = 4 + \frac{ \sqrt {1801} - 30 }{17 } $$ $$ \frac{ 17 }{ \sqrt {1801} - 30 } = \frac{ \sqrt {1801} + 30 }{53 } = 1 + \frac{ \sqrt {1801} - 23 }{53 } $$ $$ \frac{ 53 }{ \sqrt {1801} - 23 } = \frac{ \sqrt {1801} + 23 }{24 } = 2 + \frac{ \sqrt {1801} - 25 }{24 } $$ $$ \frac{ 24 }{ \sqrt {1801} - 25 } = \frac{ \sqrt {1801} + 25 }{49 } = 1 + \frac{ \sqrt {1801} - 24 }{49 } $$ $$ \frac{ 49 }{ \sqrt {1801} - 24 } = \frac{ \sqrt {1801} + 24 }{25 } = 2 + \frac{ \sqrt {1801} - 26 }{25 } $$ $$ \frac{ 25 }{ \sqrt {1801} - 26 } = \frac{ \sqrt {1801} + 26 }{45 } = 1 + \frac{ \sqrt {1801} - 19 }{45 } $$ $$ \frac{ 45 }{ \sqrt {1801} - 19 } = \frac{ \sqrt {1801} + 19 }{32 } = 1 + \frac{ \sqrt {1801} - 13 }{32 } $$ $$ \frac{ 32 }{ \sqrt {1801} - 13 } = \frac{ \sqrt {1801} + 13 }{51 } = 1 + \frac{ \sqrt {1801} - 38 }{51 } $$ $$ \frac{ 51 }{ \sqrt {1801} - 38 } = \frac{ \sqrt {1801} + 38 }{7 } = 11 + \frac{ \sqrt {1801} - 39 }{7 } $$ $$ \frac{ 7 }{ \sqrt {1801} - 39 } = \frac{ \sqrt {1801} + 39 }{40 } = 2 + \frac{ \sqrt {1801} - 41 }{40 } $$ $$ \frac{ 40 }{ \sqrt {1801} - 41 } = \frac{ \sqrt {1801} + 41 }{3 } = 27 + \frac{ \sqrt {1801} - 40 }{3 } $$ $$ \frac{ 3 }{ \sqrt {1801} - 40 } = \frac{ \sqrt {1801} + 40 }{67 } = 1 + \frac{ \sqrt {1801} - 27 }{67 } $$ $$ \frac{ 67 }{ \sqrt {1801} - 27 } = \frac{ \sqrt {1801} + 27 }{16 } = 4 + \frac{ \sqrt {1801} - 37 }{16 } $$ $$ \frac{ 16 }{ \sqrt {1801} - 37 } = \frac{ \sqrt {1801} + 37 }{27 } = 2 + \frac{ \sqrt {1801} - 17 }{27 } $$ $$ \frac{ 27 }{ \sqrt {1801} - 17 } = \frac{ \sqrt {1801} + 17 }{56 } = 1 + \frac{ \sqrt {1801} - 39 }{56 } $$ $$ \frac{ 56 }{ \sqrt {1801} - 39 } = \frac{ \sqrt {1801} + 39 }{5 } = 16 + \frac{ \sqrt {1801} - 41 }{5 } $$ $$ \frac{ 5 }{ \sqrt {1801} - 41 } = \frac{ \sqrt {1801} + 41 }{24 } = 3 + \frac{ \sqrt {1801} - 31 }{24 } $$ $$ \frac{ 24 }{ \sqrt {1801} - 31 } = \frac{ \sqrt {1801} + 31 }{35 } = 2 + \frac{ \sqrt {1801} - 39 }{35 } $$ $$ \frac{ 35 }{ \sqrt {1801} - 39 } = \frac{ \sqrt {1801} + 39 }{8 } = 10 + \frac{ \sqrt {1801} - 41 }{8 } $$ $$ \frac{ 8 }{ \sqrt {1801} - 41 } = \frac{ \sqrt {1801} + 41 }{15 } = 5 + \frac{ \sqrt {1801} - 34 }{15 } $$ $$ \frac{ 15 }{ \sqrt {1801} - 34 } = \frac{ \sqrt {1801} + 34 }{43 } = 1 + \frac{ \sqrt {1801} - 9 }{43 } $$ $$ \frac{ 43 }{ \sqrt {1801} - 9 } = \frac{ \sqrt {1801} + 9 }{40 } = 1 + \frac{ \sqrt {1801} - 31 }{40 } $$ $$ \frac{ 40 }{ \sqrt {1801} - 31 } = \frac{ \sqrt {1801} + 31 }{21 } = 3 + \frac{ \sqrt {1801} - 32 }{21 } $$ $$ \frac{ 21 }{ \sqrt {1801} - 32 } = \frac{ \sqrt {1801} + 32 }{37 } = 2 + \frac{ \sqrt {1801} - 42 }{37 } $$ $$ \frac{ 37 }{ \sqrt {1801} - 42 } = \frac{ \sqrt {1801} + 42 }{1 } = 84 + \frac{ \sqrt {1801} - 42 }{1 } $$
Solution is smallest even power of fundamental unit of form $x^2-1801$.
gp-code:
pell(d,c)=
{
Q= bnfinit('x^2-d);
fu= Q.fu[1]; print("Fundamental Unit: "fu);
N= bnfisintnorm(Q, c); print("Fundamental Solutions (Norm): "N);
for(k=1, #N, n= N[k];
for(j=0, 100,
s= lift(n*fu^j);
X= abs(polcoeff(s, 0)); Y= abs(polcoeff(s, 1));
if(Y, if(X==floor(X)&&Y==floor(Y), if(X^2-d*Y^2==c,
print("Smallest Solution (X,Y) = ("X", "Y")");
print("Smallest Power j = "j);
break(2)
)))
)
)
};
Output:
? pell(1801,1)
Fundamental Unit: Mod(14964952932154520840080376605*x - 635085521081328025318961532468, x^2 - 1801)
Fundamental Solutions (Norm): [1]
Smallest Solution (X,Y) = (806667238174283887339603999510156079461856181188670036342049, 19008049861749803182382268306718420151334269606450750222280)
Smallest Power j = 2
Verifing:
? Mod(14964952932154520840080376605*x - 635085521081328025318961532468,x^2 - 1801)^2
%1 = Mod(-19008049861749803182382268306718420151334269606450750222280*x + 806667238174283887339603999510156079461856181188670036342049, x^2 - 1801)
